# Support Vector Machine Applied to the Optimal Design of Composite Wing Panels

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- Use a limited and pre-determined number of high-fidelity finite element evaluations based on the DOE requirements;
- Use multiple objectives of continuous optimization (in opposition to integer optimization), whereby interpolation and classification objectives are built utilizing SVM models;
- Generate a Pareto frontier that discusses the contributions of mass and feasibility (buckling constraint) to the optimal design.

## 2. Structural Modeling

^{®}[22] calculates the thickness of the plate element with respect to the reference plane determined by the nodes within the element, creating a symmetrical thickness distribution profile shown in the laminate cross-section. However, the symmetric thickness distribution about the mid-plane is not consistent with the manufacturing of these panels. Thus, in order for the FE model of Figure 3 to correctly represent the as-manufactured cross-section of Figure 4, the shell elements representing the skin and stiffener’s base regions must be offset from the reference plane by a value ${t}_{sk}/2+{t}_{str}/4$, with ${t}_{sk}$ and ${t}_{str}$ being the local skin and stiffener’s thicknesses. Note that the thickness of the stiffener’s base laminate, i.e., the region connected to the skin, is ${t}_{str}/2$.

^{3}and ${t}_{ply}$ = 0.1524 mm.

#### 2.1. Load Idealization

#### 2.2. Buckling Analysis

**The first linear buckling analysis**aims at determining the critical compressive force of the panel ${F}_{cr}$, with the pre-buckling state created using a unitary force ${F}_{unit}=1\phantom{\rule{4pt}{0ex}}\mathrm{N}$ distributed in the front edge by means of a rigid element with a single master node, as illustrated in Figure 3. The critical load is calculated with ${F}_{cr}={\lambda}_{1}{F}_{unit}$, where ${\lambda}_{1}$ is the first eigenvalue of the first analysis.

**The second linear buckling analysis**evaluates the critical in-plane shear distributed force, or shear flow, of the panel ${q}_{cr}$, with the pre-buckling state created assuming a unitary in-plane shear flow ${q}_{unit}=1\phantom{\rule{4pt}{0ex}}\mathrm{N}/\mathrm{m}$. This unitary shear force is translated into nodal forces and distributed over the FE model, as illustrated in Figure 6, with the nodal forces calculated using the lengths of the representative cell in x and y directions, given by ${\ell}_{x},\text{}{\ell}_{y}$, and the number of nodes along the edges in each direction, ${{N}_{nodes}}_{x},\text{}{{N}_{nodes}}_{y}$, as per Equation (5). The critical shear force is calculated with ${q}_{cr}={\lambda}_{2}{q}_{unit}$, where ${\lambda}_{2}$ is the first eigenvalue of the second linear buckling analysis.

**The design constraint**is the margin of safety for combined shear and compression buckling. The interaction curve suggested by Kassapoglou [23] was adopted, as given in Equation (6), coupling the compression and shear effects. In this formulation, ${\sigma}_{y}$ and ${\tau}_{xy}$ correspond to the current compression and shear stresses, whereas ${\sigma}_{cr}$ and ${\tau}_{cr}$ are the critical buckling stresses for compression and shear. With the combined stress state, the panel buckles when the inequality presented in Equation (6) is not satisfied.

## 3. Methodology for the Design and Optimization

#### 3.1. Design Parameterization

- Width of the representative cell [m]: b;
- Stringer height [m]: h;
- Stringer base width [m]: ${b}_{str}$;
- Number of skin plies at ${0}^{\circ}$: ${n}_{{0}^{\circ}}^{sk}$;
- Number of skin plies at ${45}^{\circ}$: ${n}_{\pm {45}^{\circ}}^{sk}$;
- Number of skin plies at ${90}^{\circ}$: ${n}_{{90}^{\circ}}^{sk}$;
- Number of stringer plies at ${0}^{\circ}$: ${n}_{{0}^{\circ}}^{str}$;
- Number of stringer plies at ${45}^{\circ}$: ${n}_{\pm {45}^{\circ}}^{str}$;
- Number of stringer plies at ${90}^{\circ}$: ${n}_{{90}^{\circ}}^{str}$,

#### 3.2. Optimization Pipeline

- (1)
- Generate the design of experiments (DOE) ${\mathit{x}}_{\mathbf{0}}\in {\mathbb{R}}^{m\times 9}$ for the design variables;
- (2)
- Compute the failure index as per Equation (7) for each of the m FE models in the ${\mathit{x}}_{\mathbf{0}}$ designs;
- (3)
- Compute the mass index, feasibility index and feasibility classification at ${\mathit{x}}_{\mathbf{0}}$, as described next;
- (4)
- Find the optimal design ${\mathit{x}}_{\mathbf{1}}$ by the optimization of the objective (step 3) starting at ${\mathit{x}}_{\mathbf{0}}$;
- (5)
- Evaluate the FE model at the n optimal designs ${\mathit{x}}_{\mathbf{1}}$;
- (6)
- Compute the mass of the structure at the feasible designs points;
- (7)
- If new investigation is required, add information generated by ${\mathit{x}}_{\mathbf{1}}$ designs in step 5 to the initial set ${\mathit{x}}_{\mathbf{0}}$ and go to step 3; otherwise, stop.

#### 3.3. SVM Formulation

## 4. Numerical Results

#### 4.1. Metamodel Comparison

`size = 300`as the number of units in the hidden layers and

`maxit = 1000`as the maximum iterations to be learned. On the other hand, SVM used the parameters

`max_gamma = 3125`and

`min_lambda = $2.4\times {10}^{-9}$`. All three methodologies support the selection of different kernel methods, and a myriad of parameter combinations. As a result, the comparison of metamodels can lead to different results if fine-tuning is performed for each methodology.

#### 4.2. Data Analysis

#### 4.3. Sensitivity of Parameters

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Wing model, coordinate system and idealized loads at the wing root. SLZ: shear load in the z direction. BMX: bending moment about the x direction.

**Figure 2.**Schematic definition of the representative unit cell of the stiffened panel, depicted by the red dashed line. The width of the panel bay W is a given geometric parameter.

**Figure 3.**Double half-bay symmetric FE model for the representative unit cell. Force ${F}_{unit}$ represents the unitary compressive load applied for the first linear buckling analysis.

**Figure 5.**Load idealization. Top: bending moment (BMX) and shear force (SLZ) equilibrium on a single bay. Bottom: modified boom method to obtain the distributed loads of each representative cell.

**Figure 7.**(

**a**) Axial stress distribution due to bending loads; and (

**b**) shear flow distribution due to shear loads.

**Figure 17.**Feasibility error level $log\left(\right|{\mathsf{\Delta}}_{2}\left|\right)$ at several mass and feasibility values.

**Figure 18.**First DOE: optimum mass for different $\alpha $ values calculated by the FE model, after rounding the variables at the optimal points from the metamodel.

**Figure 20.**First DOE: parallel coordinate values of optimal feasible (blue) and infeasible (red) designs.

**Figure 22.**Second DOE: Pareto front of the optimum mass for different feasibility indices with $\alpha =0.5$.

**Figure 23.**First DOE: Pareto front of the optimum mass for different feasibility indices with $\alpha =0.5$.

Design Variable | Discrete Values |
---|---|

b | 105, 120, 140, 168 mm |

h | 40, 50, 60, 70, 80 mm |

${b}_{str}$ | 30, 40, 50, 60, 70 mm |

${n}_{{0}^{\circ}}^{sk}$ | $1,2,\cdots ,10$ |

${n}_{\pm {45}^{\circ}}^{sk}$ | $2,4,\cdots ,20$ |

${n}_{{90}^{\circ}}^{sk}$ | $1,2,\cdots ,10$ |

${n}_{{0}^{\circ}}^{str}$ | $1,2,\cdots ,10$ |

${n}_{\pm {45}^{\circ}}^{str}$ | $2,4,\cdots ,20$ |

${n}_{{90}^{\circ}}^{str}$ | $1,2,\cdots ,10$ |

Metamodel | Min. | 1st Qu. | Median | Mean | 3rd Qu. | Max. |
---|---|---|---|---|---|---|

SVM | $1.004\times {10}^{-5}$ | $7.243\times {10}^{-3}$ | $1.487\times {10}^{-2}$ | $2.187\times {10}^{-2}$ | $3.152\times {10}^{-2}$ | $1.717\times {10}^{-1}$ |

RBFNN | 0.00032 | 0.06771 | 0.14846 | 0.21802 | 0.28282 | 1.94824 |

MLPNN | 0.06726 | 0.59707 | 0.67307 | 0.65497 | 0.72852 | 0.86448 |

Metamodel | Min. | 1st Qu. | Median | Mean | 3rd Qu. | Max. |
---|---|---|---|---|---|---|

SVM | 0.00054 | 0.65989 | 1.58148 | 2.00729 | 2.87262 | 11.59350 |

RBFNN | 0.00025 | 0.46714 | 1.10598 | 2.05893 | 2.72801 | 24.07796 |

MLPNN | 0.00380 | 0.44705 | 1.11403 | 2.09116 | 2.75163 | 24.93674 |

i | $\mathit{\alpha}$ | $\mathit{F}\left({\mathit{x}}_{0}\right)$ | $\mathit{F}\left({\mathit{x}}_{1}\right)$ | ${\mathit{f}}_{1}\left({\mathit{x}}_{1}\right)$ | ${\mathit{f}}_{2}\left({\mathit{x}}_{1}\right)$ | ${\mathit{f}}_{3}\left({\mathit{x}}_{1}\right)$ | Time (s) |
---|---|---|---|---|---|---|---|

1 | 0.050 | 2.79 | 0.36 | 2.68 | 0.00 | 0 | 14 |

2 | 0.075 | 2.88 | 0.52 | 2.63 | 0.00 | 0 | 15 |

3 | 0.100 | 2.98 | 0.57 | 2.38 | 0.00 | 0 | 13 |

4 | 0.125 | 3.07 | 0.80 | 2.52 | 0.00 | 0 | 12 |

5 | 0.150 | 3.17 | 0.80 | 2.31 | 0.00 | 0 | 15 |

6 | 0.175 | 3.26 | 0.99 | 2.38 | 0.00 | 0 | 12 |

7 | 0.200 | 3.36 | 1.20 | 2.45 | 0.00 | 0 | 10 |

8 | 0.225 | 3.45 | 1.41 | 2.51 | 0.00 | 0 | 11 |

9 | 0.250 | 3.55 | 1.41 | 2.38 | 0.00 | 0 | 11 |

10 | 0.275 | 3.64 | 1.38 | 2.24 | 0.00 | 0 | 11 |

11 | 0.300 | 3.74 | 1.57 | 2.29 | 0.00 | 0 | 11 |

12 | 0.325 | 3.83 | 1.70 | 2.28 | 0.00 | 0 | 12 |

13 | 0.350 | 3.93 | 1.85 | 2.30 | 0.00 | 0 | 12 |

14 | 0.375 | 4.02 | 2.05 | 2.34 | 0.00 | 0 | 15 |

15 | 0.400 | 4.12 | 2.07 | 2.27 | 0.00 | 0 | 10 |

16 | 0.425 | 4.21 | 2.21 | 2.28 | 0.00 | 0 | 10 |

17 | 0.450 | 4.31 | 2.30 | 2.26 | 0.00 | 0 | 13 |

18 | 0.475 | 4.41 | 2.52 | 2.30 | 0.00 | 0 | 10 |

19 | 0.500 | 4.50 | 2.58 | 2.27 | 0.00 | 0 | 9 |

20 | 0.525 | 4.60 | 2.70 | 2.27 | 0.00 | 0 | 10 |

21 | 0.550 | 4.69 | 2.17 | 1.77 | 0.00 | 1 | 13 |

22 | 0.575 | 4.79 | 3.04 | 2.30 | 0.00 | 0 | 11 |

23 | 0.600 | 4.88 | 2.08 | 1.67 | 0.00 | 1 | 10 |

24 | 0.625 | 4.98 | 1.47 | 1.28 | 0.17 | 1 | 9 |

25 | 0.650 | 5.07 | 1.48 | 1.26 | 0.27 | 1 | 8 |

26 | 0.675 | 5.17 | 1.48 | 1.24 | 0.37 | 1 | 10 |

27 | 0.700 | 5.26 | 1.49 | 1.22 | 0.51 | 1 | 10 |

28 | 0.725 | 5.36 | 1.48 | 1.19 | 0.67 | 1 | 10 |

29 | 0.750 | 5.45 | 1.47 | 1.16 | 0.87 | 1 | 9 |

30 | 0.775 | 5.55 | 1.46 | 1.12 | 1.11 | 1 | 9 |

31 | 0.800 | 5.64 | 1.43 | 1.09 | 1.43 | 1 | 10 |

32 | 0.825 | 5.74 | 1.39 | 1.04 | 1.84 | 1 | 9 |

33 | 0.850 | 5.83 | 1.34 | 0.99 | 2.39 | 1 | 10 |

34 | 0.875 | 5.93 | 1.27 | 0.93 | 3.16 | 1 | 9 |

35 | 0.900 | 6.02 | 1.17 | 0.85 | 4.27 | 1 | 10 |

36 | 0.925 | 6.12 | 1.04 | 0.74 | 5.99 | 1 | 9 |

37 | 0.950 | 6.21 | 0.84 | 0.60 | 8.91 | 1 | 10 |

i | b | h | ${\mathit{b}}_{\mathit{s}\mathit{t}\mathit{r}}$ | ${\mathit{n}}_{{0}^{\circ}}^{\mathit{s}\mathit{k}}$ | ${\mathit{n}}_{\pm {45}^{\circ}}^{\mathit{s}\mathit{k}}$ | ${\mathit{n}}_{{90}^{\circ}}^{\mathit{s}\mathit{k}}$ | ${\mathit{n}}_{{0}^{\circ}}^{\mathit{s}\mathit{t}\mathit{r}}$ | ${\mathit{n}}_{\pm {45}^{\circ}}^{\mathit{s}\mathit{t}\mathit{r}}$ | ${\mathit{n}}_{{90}^{\circ}}^{\mathit{s}\mathit{t}\mathit{r}}$ |
---|---|---|---|---|---|---|---|---|---|

1 | 140 | 40 | 62 | 1.96 | 6.85 | 1.07 | 8.92 | 2.77 | 2.73 |

2 | 140 | 40 | 58 | 1.85 | 7.65 | 1.05 | 9.19 | 1.82 | 2.51 |

3 | 140 | 40 | 59 | 1.82 | 7.49 | 1.19 | 9.11 | 1.20 | 1.81 |

4 | 168 | 40 | 52 | 2.08 | 8.92 | 1.01 | 8.00 | 2.40 | 2.71 |

5 | 140 | 40 | 61 | 2.27 | 6.18 | 1.79 | 8.72 | 1.63 | 2.17 |

6 | 140 | 40 | 60 | 2.38 | 7.07 | 1.00 | 8.74 | 1.61 | 1.35 |

7 | 168 | 40 | 54 | 2.11 | 8.83 | 1.02 | 7.48 | 2.97 | 1.94 |

8 | 168 | 40 | 52 | 1.72 | 9.21 | 1.14 | 7.59 | 2.75 | 1.72 |

9 | 168 | 40 | 56 | 2.91 | 7.45 | 1.46 | 8.88 | 2.66 | 1.19 |

10 | 168 | 40 | 65 | 1.00 | 6.79 | 1.36 | 8.70 | 1.70 | 2.59 |

11 | 168 | 42 | 61 | 1.64 | 4.84 | 5.29 | 8.34 | 1.91 | 2.29 |

12 | 140 | 40 | 60 | 2.68 | 4.58 | 3.74 | 9.66 | 1.40 | 1.92 |

13 | 140 | 41 | 67 | 1.40 | 3.49 | 6.27 | 9.94 | 1.15 | 1.84 |

14 | 140 | 43 | 61 | 1.86 | 5.08 | 3.53 | 8.92 | 1.42 | 2.41 |

15 | 140 | 40 | 60 | 1.72 | 4.54 | 5.60 | 9.54 | 1.22 | 1.01 |

16 | 140 | 40 | 60 | 1.77 | 4.40 | 6.05 | 9.03 | 1.26 | 1.60 |

17 | 140 | 40 | 62 | 1.78 | 4.39 | 5.34 | 8.79 | 2.09 | 1.00 |

18 | 140 | 40 | 64 | 2.51 | 3.59 | 6.40 | 8.97 | 1.94 | 1.49 |

19 | 168 | 40 | 59 | 2.87 | 3.44 | 7.21 | 8.83 | 2.04 | 1.44 |

20 | 140 | 41 | 60 | 2.24 | 4.00 | 5.57 | 9.15 | 2.11 | 1.00 |

21 | 168 | 44 | 62 | 2.63 | 4.14 | 4.32 | 8.78 | 1.00 | 1.00 |

22 | 140 | 40 | 64 | 2.04 | 5.50 | 2.75 | 8.10 | 1.98 | 1.00 |

23 | 168 | 42 | 66 | 1.97 | 5.29 | 2.99 | 7.11 | 1.00 | 1.00 |

24 | 168 | 45 | 64 | 1.00 | 7.02 | 1.00 | 1.00 | 1.00 | 1.00 |

25 | 168 | 45 | 65 | 1.00 | 6.89 | 1.00 | 1.00 | 1.00 | 1.00 |

26 | 168 | 45 | 65 | 1.00 | 6.76 | 1.00 | 1.00 | 1.00 | 1.00 |

27 | 168 | 45 | 65 | 1.00 | 6.61 | 1.00 | 1.00 | 1.00 | 1.00 |

28 | 168 | 45 | 65 | 1.00 | 6.45 | 1.00 | 1.00 | 1.00 | 1.00 |

29 | 168 | 45 | 66 | 1.00 | 6.26 | 1.00 | 1.00 | 1.00 | 1.00 |

30 | 168 | 45 | 66 | 1.00 | 6.06 | 1.00 | 1.00 | 1.00 | 1.00 |

31 | 168 | 46 | 66 | 1.00 | 5.83 | 1.00 | 1.00 | 1.00 | 1.00 |

32 | 168 | 46 | 66 | 1.00 | 5.56 | 1.00 | 1.00 | 1.00 | 1.00 |

33 | 168 | 46 | 67 | 1.00 | 5.24 | 1.00 | 1.00 | 1.00 | 1.00 |

34 | 168 | 47 | 67 | 1.00 | 4.86 | 1.00 | 1.00 | 1.00 | 1.00 |

35 | 168 | 47 | 68 | 1.00 | 4.39 | 1.00 | 1.00 | 1.00 | 1.00 |

36 | 168 | 48 | 68 | 1.00 | 3.79 | 1.00 | 1.00 | 1.00 | 1.00 |

37 | 168 | 50 | 69 | 1.00 | 2.96 | 1.00 | 1.00 | 1.00 | 1.00 |

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## Share and Cite

**MDPI and ACS Style**

dos Santos, R.R.; Machado, T.G.d.P.; Castro, S.G.P.
Support Vector Machine Applied to the Optimal Design of Composite Wing Panels. *Aerospace* **2021**, *8*, 328.
https://doi.org/10.3390/aerospace8110328

**AMA Style**

dos Santos RR, Machado TGdP, Castro SGP.
Support Vector Machine Applied to the Optimal Design of Composite Wing Panels. *Aerospace*. 2021; 8(11):328.
https://doi.org/10.3390/aerospace8110328

**Chicago/Turabian Style**

dos Santos, Rogério Rodrigues, Tulio Gomes de Paula Machado, and Saullo Giovani Pereira Castro.
2021. "Support Vector Machine Applied to the Optimal Design of Composite Wing Panels" *Aerospace* 8, no. 11: 328.
https://doi.org/10.3390/aerospace8110328